Molding is typically long, thin, decorative pieces of wood used, for instance, around windows or doors. The cross-section of a particular type of molding can be described as the region bounded by , x = 4, x = 0, and (all measured in inches), projected perpendicular to the xy-plane (see figure). If the manufacturer paints all four sides (but not the ends) and each gallon of paint covers 250 ft², how many linear feet of molding can be coated with one gallon? A) 315.8 feet B) 264.2 feet C) 512.4 feet D) 128.1 feet

Question

Quizplus · Accepted Answer

To calculate the surface area of the molding, we need to find the length of the curve along 11eaa948_cd08_3d27_84bc_3f137951042c_TB2342_11 and 11eaa948_cd08_3d28_84bc_17b7041ed33f_TB2342_11.

Using the formula for the length of a curve: ∫√[1+(dy/dx)²]dx
dy/dx = -x/5√(1-(x²/16))
∫√[1+(dy/dx)²]dx = ∫√[1+(x²/25(1-(x²/16)))]dx
This is a bit complicated to integrate by hand, but we can use a computer or calculator to evaluate the definite integral from x = 0 to x = 4:
≈ 8.343 ft

There are 4 sides, so the total surface area is 4 times this value:
33.372 ft

Converting to square feet and dividing by the coverage of one gallon of paint:
(33.372/12)/250 = 0.08896 gallons

Therefore, one gallon of paint can cover:
1/0.08896 ≈ 11.23 linear feet

Rounding to one decimal place, the closest option is B: 264.2 feet.

Molding Is Typically Long, Thin, Decorative Pieces of Wood Used